Finite Group Databasebeta

← #13362382

Subgroups — #13362382

Order 1792.

1062 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89487C211yesmaximal, characteristic
2C2×#89488C211yesmaximal, characteristic, complemented
3C2×#89482C211yesmaximal, characteristic
4C2×C4×Dic28C211yesmaximal, characteristic, complemented
5C2×C4×#1132C211yesmaximal, characteristic
6#11730833C211yesmaximal, characteristic, complemented
7#11730832C211yesmaximal, characteristic
8C7×#1726C211yesmaximal, characteristic, complemented
9#11727580C211yesmaximal, characteristic
10#11727578C211yesmaximal, characteristic, complemented
11#11727570C211yesmaximal, characteristic, complemented
12#11727529C211yesmaximal, characteristic, complemented
13#11727528C211yesmaximal, characteristic
14#11727527C211yesmaximal, characteristic
15#11727525C211yesmaximal, characteristic, complemented
16C22×Dic28C2211yescharacteristic
17C22×Dic28C2211yescharacteristic
18C22×#1132C2211yescharacteristic
19C22×#1132C2211yescharacteristic
20C2×#6438C2211yescharacteristic
21C2×C4×D28C2211yescharacteristic
22C2×#6432C2211yescharacteristic
23C2×C4×Dic14C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C2×C7×#97C2211yescharacteristic
27C2×C7×#97C2211yescharacteristic
28C2×C7×#97C2211yescharacteristic
29C2×C4×C7×C8C2211yescharacteristic
30C2×C4×C7×C8C2211yescharacteristic
31C2×C4×C7×C8C2211yescharacteristic
32C7×#376C2211yescharacteristic
33C2×#6395C2211yescharacteristic
34C2×#6395C2211yescharacteristic
35C2×#6392C2211yescharacteristic
36C2×#6392C2211yescharacteristic
37C2×#6391C2211yescharacteristic
38C2×#6391C2211yescharacteristic
39C2×#6390C2211yescharacteristic
40C2×#6390C2211yescharacteristic
41C2×#6390C2211yescharacteristic
42C2×#6390C2211yescharacteristic
43C2×#6390C2211yescharacteristic
44C2×#6390C2211yescharacteristic
45#89711C2211yescharacteristic
46#89708C2211yescharacteristic
47#89707C2211yescharacteristic
48#89701C2211yescharacteristic
49#89701C2211yescharacteristic
50#89701C2211yescharacteristic
51#89488—24no—
52#89487C418yescomplemented
53#89482—24no—
54C4×Dic28—24no—
55C4×#1132—24no—
56#89442—22no—
57#89440—22no—
58#89425—22no—
59#89425—22no—
60#21703—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×Dic14C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×(C2×D7)⋊C4—22no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—22no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×Dic28—22no—
89C2×Dic28—22no—
90C2×Dic28—22no—
91C2×Dic28—22no—
92C2×Dic28C2×C414yescomplemented
93C2×Dic28C2×C414yescomplemented
94C2×#1132—22no—
95C2×#1132—22no—
96C2×#1132—22no—
97C2×#1132—22no—
98C2×#1132C2×C414yes—
99C2×#1132C2×C414yes—
100#6438—22no—
101#6438C2×C414yes—
102C4×D28—22no—
103C4×D28—22no—
104#6432—22no—
105#6432C2×C414yescomplemented
106C4×Dic14—22no—
107C4×Dic14—22no—
108C4×Dic14—22no—
109C4×Dic14—22no—
110C4×Dic14—22no—
111C4×Dic14—22no—
112C7×#97—22no—
113C7×#97C2×C414yescomplemented
114C7×#97D414yes—
115C4×C7×C8—22no—
116C4×C7×C8—22no—
117C4×C7×C8D414yes—
118C7×#95D412yes—
119C7×#95D412yes—
120C7⋊#95—22no—
121C7⋊#95—22no—
122C7⋊#95—22no—
123C7⋊#95—22no—
124#6395—22no—
125#6395—22no—
126#6392—22no—
127#6392—22no—
128#6391—22no—
129#6391—22no—
130#6390—22no—
131#6390—22no—
132#6390—22no—
133#6390—22no—
134#6390—22no—
135#6390—22no—
136C2×#497—71no—
137C2×#441—71no—
138C2×#440—71no—
139C2×C4×Q16—71no—
140C2×C4×SD16—71no—
141#2327—71no—
142#2326—71no—
143#1973—71no—
144#1971—71no—
145#1969—71no—
146#1968—71no—
147#1958—71no—
148#1956—71no—
149#1847—71no—
150#1726D711yescharacteristic, complemented
151C23×D7—21no—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C2×D28—21no—
165C2×D28—21no—
166C2×D28—21no—
167C2×D28—21no—
168C2×D28—21no—
169C2×D28C22×C412yes—
170C2×C4×D7—42no—
171C2×C4×D7—82no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14—21no—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14C22×C412yes—
188C2×Dic14C22×C412yes—
189C2×Dic14C22×C412yes—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8C22×C412yes—
195C2×C7×C8C22×C412yes—
196C2×C7×C8C2×D412yes—
197C2×C7×C8C2×D412yes—
198C2×C7×C8C2×D412yes—
199C2×C7×C8C2×D412yes—
200C2×C7×C8#4112yes—
201C2×C7×C8#4112yes—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7C22×C412yes—
205C42×C7C2×D412yes—
206C42×C7C2×D412yes—
207C42×C7#4112yes—
208(C2×D7)⋊C4—28no—
209(C2×D7)⋊C4—42no—
210(C2×D7)⋊C4—82no—
211C4⋊Dic7—28no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—82no—
221Dic7⋊C4—42no—
222Dic7⋊C4—42no—
223Dic7⋊C4—42no—
224C4×Dic7—42no—
225C4×Dic7—42no—
226C4×Dic7—42no—
227Dic28—28no—
228Dic28—28no—
229#1132—28no—
230#1132—28no—
231C22×Q16—71no—
232C22×Q16—71no—
233C22×SD16—71no—
234C22×SD16—71no—
235C2×C4⋊Q8—71no—
236C2×#124—71no—
237C2×C4×Q8—71no—
238C2×C4×Q8—71no—
239C2×C4×Q8—71no—
240C2×C4×D4—71no—
241#497—78no—
242#441—144no—
243#440—144no—
244C4×Q16—144no—
245C4×SD16—144no—
246C2×#107—71no—
247C2×#107—71no—
248C2×#106—71no—
249C2×#106—71no—
250C2×#103—71no—
251C2×#103—71no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#102—71no—
257C2×#102—71no—
258C2×#97C2×D711yescharacteristic
259C2×#97C2×D711yescharacteristic
260C2×#97C2×D711yescharacteristic
261C2×C4×C8C2×D711yescharacteristic
262C2×C4×C8C2×D711yescharacteristic
263C2×C4×C8C2×D711yescharacteristic
264#391—71no—
265#391—71no—
266#391—71no—
267#390—71no—
268#388—142no—
269#387—142no—
270#387—142no—
271#385—142no—
272#384—71no—
273#382—71no—
274#376C2×D711yescharacteristic
275C23×C7C22×D411yescharacteristic
276C22×D7—22no—
277C22×D7—41no—
278C22×D7—41no—
279C22×D7—42no—
280C22×D7—42no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7C4×D414yes—
289C2×C4×C7#12712yes—
290C2×C4×C7#12712yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312C2×Dic7—42no—
313C2×Dic7—42no—
314D28—22no—
315D28—22no—
316D28—41no—
317D28—41no—
318C4×D7—84no—
319Dic14—22no—
320Dic14—22no—
321Dic14—22no—
322Dic14—22no—
323Dic14—22no—
324Dic14—22no—
325Dic14—41no—
326Dic14—41no—
327Dic14—41no—
328Dic14—41no—
329Dic14—41no—
330Dic14—41no—
331C7×C8—22no—
332C7×C8—22no—
333C7×C8C4×D414yes—
334C7×C8C4×D414yes—
335C22×Q8—71no—
336C22×Q8—71no—
337C22×Q8—71no—
338C22×D4—71no—
339C23×C4—141no—
340C2×Q16—74no—
341C2×Q16—74no—
342C2×Q16—142no—
343C2×Q16—142no—
344C2×Q16—142no—
345C2×Q16—142no—
346C2×SD16—74no—
347C2×SD16—74no—
348C2×SD16—142no—
349C2×SD16—142no—
350C2×SD16—142no—
351C2×SD16—142no—
352C22×C8C22×D711yescharacteristic
353C22×C8C22×D711yescharacteristic
354C22×C8C22×D711yescharacteristic
355C22×C8C22×D711yescharacteristic
356C4⋊Q8—74no—
357C4⋊Q8—142no—
358#124—74no—
359#124—142no—
360C4×Q8—142no—
361C4×Q8—142no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×D4—142no—
367C4×D4—142no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—142no—
377C2×C22⋊C4—142no—
378C2×C22⋊C4—141no—
379C2×C22⋊C4—142no—
380C2×C42—141no—
381C2×C42—141no—
382C2×C42—141no—
383C2×C42C22×D711yescharacteristic
384C2×C42C22×D711yescharacteristic
385C2×C42C22×D711yescharacteristic
386#107—142no—
387#107—142no—
388#106—142no—
389#106—142no—
390#103—142no—
391#103—142no—
392#103—142no—
393#103—142no—
394#103—142no—
395#103—142no—
396#102—142no—
397#102—142no—
398#97—22no—
399#97C4×D714yescomplemented
400#97D2814yes—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8D2814yes—
404#95—142no—
405#95—142no—
406#95—142no—
407#95—142no—
408#95D2812yes—
409#95D2812yes—
410C22×C7C2×C4×D411yescharacteristic
411C22×C7C2×#12711yescharacteristic
412C22×C7#53211yescharacteristic
413C22×C7#57712yes—
414C22×C7#57812yes—
415C2×D7—42no—
416C2×D7—44no—
417C2×D7—81no—
418C2×D7—81no—
419C2×D7—82no—
420C4×C7—22no—
421C4×C7—22no—
422C4×C7—24no—
423C4×C7—42no—
424C4×C7C2×C4×D411yescharacteristic
425C4×C7C2×C4×D411yescharacteristic
426C4×C7#53211yescharacteristic
427C4×C7#53211yescharacteristic
428Dic7—42no—
429Dic7—42no—
430Dic7—42no—
431Dic7—81no—
432Dic7—81no—
433Dic7—81no—
434Dic7—81no—
435C24—141no—
436C2×Q8—72no—
437C2×Q8—72no—
438C2×Q8—72no—
439C2×Q8—141no—
440C2×Q8—141no—
441C2×Q8—141no—
442C2×Q8—141no—
443C2×Q8—141no—
444C2×Q8—141no—
445C2×Q8—141no—
446C2×Q8—141no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×Q8—141no—
450C2×Q8—141no—
451C2×Q8—141no—
452C2×Q8—141no—
453C2×Q8—141no—
454C2×D4—72no—
455C2×D4—141no—
456C2×D4—141no—
457C2×D4—141no—
458C2×D4—141no—
459C2×D4—141no—
460C22×C4—141no—
461C22×C4—141no—
462C22×C4—141no—
463C22×C4—141no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—282no—
468C22×C4—562no—
469C22×C4C2×D2812yes—
470C22×C4C2×D2812yes—
471C22×C4C2×D2811yescharacteristic
472C22×C4C2×D2811yescharacteristic
473C22×C4C23×D711yescharacteristic
474Q16—148no—
475Q16—148no—
476SD16—148no—
477SD16—148no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8C2×C4×D712yes—
483C2×C8C2×C4×D712yes—
484C2×C8C2×D2812yes—
485C2×C8C2×D2812yes—
486C2×C8C2×D2812yes—
487C2×C8C2×D2812yes—
488C2×C8#115712yes—
489C2×C8#115712yes—
490C4⋊C4—148no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—282no—
500C4⋊C4—282no—
501C4⋊C4—282no—
502C4⋊C4—562no—
503C22⋊C4—148no—
504C22⋊C4—282no—
505C22⋊C4—562no—
506C42—21no—
507C42—21no—
508C42—282no—
509C42—282no—
510C42—282no—
511C42C2×C4×D712yes—
512C42C2×D2812yes—
513C42C2×D2812yes—
514C42#115712yes—
515C2×C7C4×#12711yescharacteristic
516C2×C7#294712yes—
517C2×C7#315411yescharacteristic
518C2×C7#315611yescharacteristic
519C2×C7#317812yes—
520D7—82no—
521C23—142no—
522C23—281no—
523C23—281no—
524C23—282no—
525C23—282no—
526C23C22×D2811yescharacteristic
527Q8—142no—
528Q8—142no—
529Q8—142no—
530Q8—142no—
531Q8—142no—
532Q8—142no—
533Q8—281no—
534Q8—281no—
535Q8—281no—
536Q8—281no—
537Q8—281no—
538Q8—281no—
539D4—142no—
540D4—142no—
541D4—281no—
542D4—281no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4—282no—
566C2×C4—282no—
567C2×C4—564no—
568C2×C4C4×D2814yes—
569C2×C4#643712yes—
570C2×C4#643712yes—
571C2×C4C22×C4×D711yescharacteristic
572C2×C4C22×D2811yescharacteristic
573C2×C4C22×D2811yescharacteristic
574C2×C4C2×#115711yescharacteristic
575C2×C4C2×#115711yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C8—22no—
578C8—22no—
579C8C4×D2814yes—
580C8C4×D2814yes—
581C7#2170311yescharacteristic, complemented, Sylow
582C22—282no—
583C22—284no—
584C22—561no—
585C22—562no—
586C22—561no—
587C22C2×C4×D2811yescharacteristic
588C22C2×#643711yescharacteristic
589C22#9016811yescharacteristic
590C22#9044212yes—
591C22#9044412yes—
592C4—22no—
593C4—24no—
594C4—22no—
595C4—42no—
596C4—282no—
597C4—282no—
598C4—282no—
599C4—561no—
600C4—561no—
601C4—561no—
602C4—561no—
603C4C2×C4×D2811yescharacteristic
604C4C2×C4×D2811yescharacteristic
605C4#9016811yescharacteristic
606C4#9016811yescharacteristic
607C2—562no—
608C2C4×#643711yescharacteristic
609C2#1173600911yescharacteristic
610C2#1173601512yes—
611C2#1173601612yes—
612C2#1173602111yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1726C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1726D7
hypocenterC7#21703
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #1726

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#89488C2semidirect
4. C2×C4×Dic28C2semidirect
6. #11730833C2semidirect
8. C7×#1726C2semidirect
10. #11727578C2semidirect
11. #11727570C2semidirect
12. #11727529C2semidirect
15. #11727525C2semidirect
52. #89487C4semidirect
92. C2×Dic28C2×C4semidirect
93. C2×Dic28C2×C4semidirect
105. #6432C2×C4semidirect
113. C7×#97C2×C4semidirect
150. #1726D7semidirect
399. #97C4×D7semidirect
581. C7#21703semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#894872
C2×#894882
C2×#894822
C2×C4×Dic282
C2×C4×#11322
#117308332
#117308322
C7×#17262
#117275802
#117275782
#117275702
#117275292
#117275282
#117275272
#117275252
#217037